Sporadic group

In group theory, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.

A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families, plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group,[1] in which case the sporadic groups number 27.

The monster group is the largest of the sporadic groups and contains all but six of the other sporadic groups as subgroups or subquotients.

Names of the sporadic groups

Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is:

The Tits group T is sometimes also regarded as a sporadic group (it is almost but not strictly a group of Lie type), which is why in some sources the number of sporadic groups is given as 27 instead of 26.[2] In some other sources, the Tits group is regarded as neither sporadic nor of Lie type.[3] Anyway, it is the (n=0)-member 2F4(2)′ of the infinite family of commutator groups 2F4(22n+1)′, all of them finite simple groups. For n>0 they coincide with the groups of Lie type 2F4(22n+1). But for n=0, the derived subgroup 2F4(2)′, called Tits group, has an index 2 in the group 2F4(2) of Lie type.

Matrix representations over finite fields for all the sporadic groups have been constructed.

The earliest use of the term "sporadic group" may be Burnside (1911, p. 504, note N) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received".

The diagram at right is based on Ronan (2006). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.

Organization

Of the 26 sporadic groups, 20 can be seen inside the Monster group as subgroups or quotients of subgroups (sections).

I. Pariah

The six exceptions are J1, J3, J4, O'N, Ru and Ly. These six are sometimes known as the pariahs.

II. Happy Family

The remaining twenty have been called the Happy Family by Robert Griess, and can be organized into three generations.

First generation (5 groups): the Mathieu groups

Mn for n = 11, 12, 22, 23 and 24 are multiply transitive permutation groups on n points. They are all subgroups of M24, which is a permutation group on 24 points.

Second generation (7 groups): the Leech lattice

All the subquotients of the automorphism group of a lattice in 24 dimensions called the Leech lattice:

  • Co1 is the quotient of the automorphism group by its center {±1}
  • Co2 is the stabilizer of a type 2 (i.e., length 2) vector
  • Co3 is the stabilizer of a type 3 (i.e., length 6) vector
  • Suz is the group of automorphisms preserving a complex structure (modulo its center)
  • McL is the stabilizer of a type 2-2-3 triangle
  • HS is the stabilizer of a type 2-3-3 triangle
  • J2 is the group of automorphisms preserving a quaternionic structure (modulo its center).

Third generation (8 groups): other subgroups of the Monster

Consists of subgroups which are closely related to the Monster group M:

  • B or F2 has a double cover which is the centralizer of an element of order 2 in M
  • Fi24 has a triple cover which is the centralizer of an element of order 3 in M (in conjugacy class "3A")
  • Fi23 is a subgroup of Fi24
  • Fi22 has a double cover which is a subgroup of Fi23
  • The product of Th = F3 and a group of order 3 is the centralizer of an element of order 3 in M (in conjugacy class "3C")
  • The product of HN = F5 and a group of order 5 is the centralizer of an element of order 5 in M
  • The product of He = F7 and a group of order 7 is the centralizer of an element of order 7 in M.
  • Finally, the Monster group itself is considered to be in this generation.

(This series continues further: the product of M12 and a group of order 11 is the centralizer of an element of order 11 in M.)

The Tits group also belongs in this generation: there is a subgroup S4 ×2F4(2) normalising a 2C2 subgroup of B, giving rise to a subgroup 2·S4 ×2F4(2) normalising a certain Q8 subgroup of the Monster. 2F4(2) is also a subgroup of the Fischer groups Fi22, Fi23 and Fi24, and of the Baby Monster B. 2F4(2) is also a subgroup of the (pariah) Rudvalis group Ru, and has no involvements in sporadic simple groups except the containments we have already mentioned.

Table of the sporadic group orders

GroupGenerationOrder (sequence A001228 in the OEIS)1SFFactorized orderStandard generators triple (a, b, ab)[4][5][2]Further conditions
F1 or Mthird8080174247945128758864599049617107
57005754368000000000
≈ 8×1053246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 712A, 3B, 29none
F2 or Bthird4154781481226426191177580544000000 ≈ 4×1033 241 · 313 · 56 · 72 · 11 · 13 · 17 · 19 · 23 · 31 · 472C, 3A, 55
Fi24' or F3+third1255205709190661721292800≈ 1×1024221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 292A, 3E, 29
Fi23third4089470473293004800≈ 4×1018218 · 313 · 52 · 7 · 11 · 13 · 17 · 232B, 3D, 28none
Fi22third64561751654400≈ 6×1013217 · 39 · 52 · 7 · 11 · 132A, 13, 11
F3 or Ththird90745943887872000≈ 9×1016215 · 310 · 53 · 72 · 13 · 19 · 312, 3A, 19none
Lypariah51765179004000000≈ 5×101628 · 37 · 56 · 7 · 11 · 31 · 37 · 672, 5A, 14
F5 or HNthird273030912000000≈ 3×1014214 · 36 · 56 · 7 · 11 · 192A, 3B, 22
Co1second4157776806543360000≈ 4×1018221 · 39 · 54 · 72 · 11 · 13 · 232B, 3C, 40none
Co2second42305421312000≈ 4×1013218 · 36 · 53 · 7 · 11 · 232A, 5A, 28none
Co3second495766656000≈ 5×1011210 · 37 · 53 · 7 · 11 · 232A, 7C, 17none
O'Npariah460815505920≈ 5×101129 · 34 · 5 · 73 · 11 · 19 · 312A, 4A, 11none
Suzsecond448345497600≈ 4×1011213 · 37 · 52 · 7 · 11 · 132B, 3B, 13
Rupariah145926144000≈ 1×1011214 · 33 · 53 · 7 · 13 · 292B, 4A, 13none
F7 or Hethird4030387200≈ 4×109210 · 33 · 52 · 73 · 172A, 7C, 17none
McLsecond898128000≈ 9×10827 · 36 · 53 · 7 · 112A, 5A, 11
HSsecond44352000≈ 4×10729 · 32 · 53 · 7 · 112A, 5A, 11none
J4pariah86775571046077562880≈ 9×1019221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 432A, 4A, 37
J3 or HJMpariah50232960≈ 5×10727 · 35 · 5 · 17 · 192A, 3A, 19
J2 or HJsecond604800≈ 6×10527 · 33 · 52 · 72B, 3B, 7
J1pariah175560≈ 2×10523 · 3 · 5 · 7 · 11 · 192, 3, 7
M24first244823040≈ 2×108210 · 33 · 5 · 7 · 11 · 232B, 3A, 23
M23first10200960≈ 1×10727 · 32 · 5 · 7 · 11 · 232, 4, 23
M22first443520≈ 4×10527 · 32 · 5 · 7 · 112A, 4A, 11
M12first95040≈ 1×10526 · 33 · 5 · 112B, 3B, 11none
M11first7920≈ 8×10324 · 32 · 5 · 112, 4, 11

References

  1. For example, by John Conway.
  2. Wilson RA, Parker RA, Nickerson SJ, Bray JN (1999). "Atlas: Sporadic Groups".
  3. In Eric W. Weisstein „Tits Group“ From MathWorld--A Wolfram Web Resource the Tits group is given the attribute sporadic, whereas in Eric W. Weisstein „Sporadic Group“ From MathWorld--A Wolfram Web Resource, however, the Tits group is NOT listed among the 26. Both sources checked on 2018-05-26.
  4. Wilson RA (1998). "An Atlas of Sporadic Group Representations" (PDF).
  5. Nickerson SJ, Wilson RA (2000). "Semi-Presentations for the Sporadic Simple Groups".
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